A122928 Coefficients of a q-series inspired by Andrews and Ramanujan.
1, 1, 2, 3, 5, 8, 12, 18, 26, 37, 52, 72, 99, 134, 180, 240, 317, 416, 542, 702, 904, 1158, 1476, 1872, 2364, 2973, 3724, 4647, 5778, 7160, 8844, 10890, 13370, 16368, 19984, 24336, 29561, 35822, 43308, 52242, 62884, 75536, 90552, 108342, 129384, 154232
Offset: 0
Keywords
Links
- G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Crossrefs
Cf. A098693(n)=a(n) if n>0.
Programs
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Mathematica
nmax = 100; CoefficientList[Series[Product[(1-x^(12*k))*(1+x^(12*k-5))*(1+x^(12*k-7))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *) -
PARI
{a(n)=if(n<1, n==0, polcoeff( sum(k=1, sqrtint(n), x^k^2/(1+x^k)* prod(i=1, k, (1+x^i)^2/(1-x^(2*i-1))/(1-x^(2*i)), 1+x*O(x^(n-k^2)))), n))}
Formula
Euler transform of period 24 sequence [ 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, ...].
Given g.f. A(x), then B(x)=A(x)^2-A(x) satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+6*u)*v*(1+2*v)-u^2.
G.f.: {Sum_{k} q^(6k^2-k) }/{Sum_{k} (-1)^k q^((3k^2-k)/2) }.
G.f.: Product_{k>0} (1-q^(12k))(1+q^(12k-5))(1+q^(12k-7))/(1-q^k).
G.f.: 1+Sum_{k>0} Prod[i=1..k, (1+q^i)^2]*(1+q^k)*q^(k^2) /{(1-q)(1-q^2)...(1-q^(2k))}.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015